A Capital Asset Pricing Model’s (CAPM’s) Beta Estimation in the Presence of Normality and Non-normality Assumptions

This study describes the approach for estimating the beta-risk of the Capital Asset Price Model (CAPM) when the normality (Gaussian) assumption of both the error term and the excess return on an asset holds, and also when their normality assumption is violated or failed due to outliers or excessive skewness and excessive kurtosis. The student-t distribution was used as an alternative distribution to capture these anomalies. The monthly All-share Index (ASI) of 12 crucial Market Portfolios / Sectors derived from Nigeria Stock Exchange (NSE) were subjected to both the Gaussian error innovation and Student-t error innovation in this study. However, it was noted that estimates of portfolios’ beta-risk and its standard error for Gaussian and student-t were approximately the same when the sector follows a normal distribution while the standard errors of portfolio beta-risk estimates will be smaller under student-t innovation than that of Gaussian innovation when the sector does not follow normal distribution due to these anomalies. Furthermore, it was discovered that building & construction, manufacturing, quarry & mining, communication, transportation, education and utilities sectors have been having lower volatility, that is, in boosting the economy over the last 15 years.


Introduction
Risk identification and quantification are two of the major recipes to managing risk associated with business decisions. Scholars like (Sharpe, 1964;Markiwitz, 1965 andRoss, 1976) came up with theories and techniques for managing risks and uncertainty. One of the early risk management models is the Capital Asset Price Model (CAPM) which has remained a principal ornament for modeling modern financial economics. The origin of the model was arguably introduced by Treynor (1965) while the likes of Sharpe (1964) and Lintner (1965) and Markowitz (1965) made some independent contributions to furthering the concepts of the model. A close variation to the CAPM is the arbitrage pricing theory propounded by Ross (1976) which believes that the expected return of a financial instrument is a function of many macroeconomic variables with each of the variables having their own level of probability, (beta) coefficient (β).
The concept of CAPM which bothers more on systematic or market risk is based on some assumptions upon which the model can guarantee optimal risk measurement and risk-return rate on investment securities and portfolios. The systematic risk is a non-diversifiable for every investor regardless of the market or sector within an economy. The systematic risk is unavoidable, it is a risk that is common to all investors unlike unsystematic risk that is associated with a particular market or sector. The unsystematic risk is avoidable when you move to a different market, sector or economy. The bedrock of the model is on certain economic and non-economic assumptions. The assumptions include existence of rational and risk-averse investors, availability of huge investment opportunities, cost-less information, tax-less economy, free market entry and exit, homogenous expectations by all market players, absence of information asymmetry and price sacrosanct to the extent that no one can influence asset prices (Cheema, 2010;Bouchaddekh, Bouri and Kefi, 2014;Myers and Turnbull, 1977). There is no doubt that none of the assumptions is in consonance with reality and this has accounted for the major pitfall of the model as it has not been tested with live situations.
However, in spite of the supposedly impracticability of the model in live situations where information is free, taxes are paid, heterogeneous expectations and high presence of information asymmetry in the market, the contributions of CAPM Model to developing other risk measurement models that are more sensitive to economic realities cannot be overemphasized. This study has therefore decided to estimate the risk-free rate (B) in a risk-less market where the CAPM underlying assumptions subsist as well as under the reality of economic situations where the CAMP assumptions become a non-normality.
This study would shift the frontier of academic knowledge as it concerns the theories of asset pricing under the normal and non-normal (reality economics) assumptions while it would also add to theoretical references for academic researches. For investment analysts, the study would assist in broadening their knowledge in asset pricing under different assumptions and economic scenarios.

Literature Review
The alternative description of CAPM as defined by Markowitz (1965) is the Portfolio Model. The essence of the model according to Francioni and Schwartz, 2017;Gencay, Selcuk and Whitcher, 2005;Santis, 2010) is that it provides a platform for risk measurement while also establishing the correlation between future market risk and return. The essence of the model is the relationship between expected risk and return. In fact, the model according to Demircioglu (2015), and Pamane and Vikpossi (2014) describes a direct relationship between market expected return and risks. The practicability of the model to support reality has not been supported with many empirical evidences and the model is best described as a theoretical exercise because of its many unrealistic assumptions (Nyangara, Nyangara, et al., 2016;Ward and Muller, 2013;Fama and French, 2004). According to Leonard et al. (2012), he defined the model as one that gives a precise prediction of the relationship between the risk of an asset or stock and its expected return. He stressed further that the two vital functions of the model are to provide a benchmark rate of return for evaluating possible investments and one that helps in educational guess as to the expected return on assets that have not yet been traded in the marketplace.
It is rational for every investor to desire a commensurate rate of return for every element of risk taken (Dawson, 2014; Aduda and Muimi, 2011) but determining the return under realistic situations (risk presence) as opposed to the assumptions of the model (risk absence) remains a mirage (Mullins, 1982). The model according to Mullins (1982) is of good use for estimating costs of capital (equity) especially when it is used in conjunction with other risk-return estimation techniques. In fact, it is a preferred method in estimating the cost of equity than the usual dividend growth model because of its ability to emphasize on the company's systematic risks (Perold, 2004). In the stock markets, Lee, Cheng and Chong (2016) argued that investment managers engages CAPM to determine stock returns while at the same time diversifying their investment portfolios to eliminate or reduce the unsystematic risks for the purpose of maximizing profitability and shareholders' wealth. In a similar vein, Nwani (2015) affirmed what Miller (1999) stated that CAPM through its empirical evidence has been contributing immensely to the development of finance by providing unflinching insights into the form of risk involved in an asset or stock pricing. He elaborated that CAPM assumes that the expected return on any asset/stock is positively and linearly related to its market beta-risk (systematic risk), which according to the CAPM is the only relevant measure of undiversifiable risk of the asset, so the elucidating power of the CAPM's systematic risk (also known as beta risk) and abnormal return cannot be jettisoned in offering significant explanation to variations in assets/stocks in various equity market. The CAPM provides the required return based on the perceived level of systematic risk of an investment. Furthermore, it is clear that all potential return arises with risk and that low-risk (low levels of uncertainty) associated with low potential returns while high risk goes along with high potential risk. The two risks associated with the CAPM are; the unsystematic risk and the systematic risk otherwise known as the Market beta-risk or the non-diversifiable risk. The former can be eliminated through diversification while the latter is the market risk and cannot be eliminated via diversification due to factors like Interest rates, recession, inflation, unstable energy prices, changes of policies, political unstable, business cycle etc.
The Beta-risk (market risk) of the CAPM has been the index used by researchers in classifying the risk associated with an asset/stock to know whether the beta risk is aggressive, tracks the market, conservative, independent or perfectly hedged. This risk has always been estimated using cross-sectional regression analysis via Ordinary Least Square (OLS) where both the dependent variable (the excess return on an asset) and the error term are normally distributed. Jensen et al. (1972) in their study indicated that the expected excess return on an asset might not be strictly proportional to its systematic risk, they are of sufficient belief that cross-sectional estimates and analysis of the beta factor might be subjected to measurement bias error due to the traditional form of the model (that is, the joint probability distribution of all security returns). In a similar study, Oke (2013) applied the CAPM to the Nigerian stock market using weekly stock returns of 110 companies from the Nigerian Stock Exchange (NSE). To enhance the precision of the beta risk estimates and statistical problems that might arise from measurement errors in individual beta estimates, combined the securities with portfolios such that the results will invalidate the CAPM's prediction that higher beta risk is associated with a higher level of return and the abnormal return should equal zero when estimating Security Market Line (SML). Also, Osamwonyi and Asein (2012) examined the market risk defined by CAPM as an explanatory variable for security returns in most capitalized firms in the Nigeria Capital Market. His findings confirmed a positive linear relationship between market betas and security returns for the selected Nigerian firms after subjecting the CAPM to Independent and identically normal distribution. In a similar vein, Nwani (2015) subjected the UK Stock Returns to CAPM beta risk via the OLS cross-sectional regression analysis and deduced that the stock returns in the UK equity market are not significantly sensitive to the systematic (market) risk. Like Nwani (2015), Tumala and Yaya (2015) came up with an alternative method of estimating beta risk using Logistic Smooth Threshold Model (LSTM). They examined the Nigeria market Sector returns using the LSTM in order to conquer the problem of high positive skewness coefficients and high negative skewness coefficients characterized by some sector equity returns. They discovered that Petroleum, Finance, and Food and Beverages sector equities yielded higher investment risk within their study period.
The aforesaid bias measurement in the systematic (market risk) is always related to the violation of normality assumption(s) in the error term and excess return on an asset of the CAPM. These assumptions violation might be due to excessive kurtosis or excessive skewness (that is, non bell-shaped distributions), outliers (due to recession, hike in exchanges, policy changes etc.). Subjecting the CAPM to Gaussian distribution, the assumption violations will surely distort the estimates of the beta risk (market risk), so, a more robust distribution is needed in capturing these violations. The Student-t distribution theoretical framework will be used in this research work as an alternative distribution in the presence of assumption(s) violation. Hurn et al. (2015) and Oke (2013) provided a coherent and updated framework by defining beta-risk (systematic risk) that can be derived from CAPM defined by ( (Sharpe, 1964), (Treynor, 1965) and (Mossin, 1966 The slope parameter β represents the asset's beta risk. The intercept parameter α represents the abnormal return to the asset over and above the asset's exposure to the excess return on the market.

Theoretical Analysis
This can be re-written as

Estimating of the Beta-Risk under the Normality Assumption (Gaussian innovation)
Given the normal distribution as; The distribution of t r with parameter vector is The maximum likelihood estimator of { } To get the variance-covariance for ɵ η , otherwise known as Hessian matrix.
H η conforms with the negativity of the second derivative of an estimable parameter via maximum likelihood. In practice, the return distributions for stocks and assets are not symmetric instead assets do exhibit fat tails (higher kurtosis or higher skewness and outliers i.e. extreme positive values or extreme negative values), but assets do possess extreme positive values. These extreme values can come to play due to change in policies and implementation, recession, hike in exchange rates, unforeseen circumstances etc. in a country. These exhibition of fat tails and outliers by stocks and assets do affect the assumptions of the Independently and Identically Normal Distribution of the error term of the CAPM model which definitely result in distorting the estimates of beta-risk (non-diversifiable risk) ( 1 ( ) β ) and the abnormal return 0 ( ) β . The ideal solution is to replace the normality assumption with a fat-tailed distribution (student-t-distribution) that will capture and accommodate outliers and at the same time reduces the distortion of the estimates of beta-risk in the presence of outlier(s). So, a robust version of the CAPM that will replace the normal distribution with standardized mean zero and variance is needed.
The parameter " " v is the degree of freedom parameter which captures the effects outliers in the tails of the distribution.

Estimating of the Beta-Risk Under the Assumption Violation (Student-t Innovation)
Felipe et al (2004) used an EM algorithm in estimating the variance in student-t-distribution via E-step and M-step. The same Iterative procedure will be employed but for estimating . For any noted change-point detection.
We considered , , , The distribution in terms of t y gives To derive the maximum likelihood estimator of Since equation (6) for log-likelihood is a non-linear in the parameters that has no close-form an iterative solution would be adopted to compute ɵ η .
Equation (6) can be re-written as Where, So, equation (2) can be re-written as While the associated score functions are given by, . So, developing estimates procedure using the EM algorithm, the observed log-likelihood in function equation (3) becomes   2  0  1  1  1 log ( ) ( / ; ) The ( ) This based on the following weights function, Step: Using the estimates to be obtained at E-step likelihood estimates to obtain ( ) The algorithm proceeds between E and M step until the sequence ( ) r η converges. It is to be noted that the degree of freedom will be 2 v N = − .

Experimental Work
The data used in this research work was the monthly All-share Index (ASI) of Market Portfolios / Sectors indexes derived from the Nigeria Stock Exchange (NSE) from 1:2000 to 12:2015. Twelve crucial Sectors were considered based on the constant supply of their indexes by NSE, impact on the Nigerian economy and how crucial to the Nigeria market. These Sectors are agriculture, building and construction, wholesale and retail, manufacturing, quarry & mining, petrol & gas, transportation, communication, education, health, utilities, finance and insurance. It was noted that from Figure 1  From the estimated coefficients of beta-risk (systematic risk) in Table 2 below, it was noted that the beta-risk estimates for both the Gaussian error and student-t error innovations for Agricultural sector and Financial & Insurance sector were valued as (1.0868 and 1.4390) and (1.0868 and 1.2220) respectively ( 1 β 's > 1), this implies that the two sectors were nimble and operative, that is, the two sectors have been in a state of working order. In other words, the security's price in the two sectors seemed theoretically more volatile than the market (economy) around 8% and 50% respectively with the possibility of a higher rate of return over 15  β 's < 1). It seemed that the security's price in the seven sectors theoretically are less volatile than the market (economy) around 41%, 31%, 38%, 78%, 2%, 41%, 70% and 4% respectively, that is, the fund's excess return is anticipated to underachieve the benchmark by these percentages aforestated in boasting the economy (markets) and perform better by these percentages during receding economy (markets). In others, their systematic risks are fairly efficient in explaining the relationship between the economy risk and return.
It was noted that the estimated standard errors for the non-diversifiable risks in Table 2 for student-t error innovation for the seemed non-normally distributed sectors of Wholesale & Retail, Petrol & Gas, Health, and Finance ( 0.0124, 0.9539, 0.0579, 0.1311) respectively are smaller than that of the Gaussian error innovation (0.0114, 0.8832, 0.0379, 0.0100 ) respectively. This was due to the fact that normal error type model was fitted to non-normality error innovation type model. Also, the Alkaike Information Criteria (AIC) values were also smaller in student-t innovation compared to the Gaussian innovation in these seven sectors, meaning these four sectors had been fluctuating over the past 15 years. It was found that not in all cases that the error term in the CAPM model for estimating the beta-risk and the abnormal return is independently, identically and normally distributed. Subjecting the error term to student-t distribution encompasses and accommodates both fat-tailed and bell-shaped distributed variables.
The student-t error innovation approximately gives the same estimate and standard error of such estimate if the sectors/assets are independently, identically and normally distributed while subjecting the variable under consideration to Gaussian whereas it follows independent and identical student-t distribution, the estimates under student-t will the more stable. Student-t approaches the true parameter with a smaller standard error compared to indexes of the estimates from normally distributed error innovation.